## Equilibrium programming in Hilbert spaces.(English)Zbl 1109.90079

Given a Hilbert space $$\mathcal{H}$$, a closed convex subset $$K$$ of $$\mathcal{H}$$ and a countable family of functions $$F_{i}\colon K^2\to R$$ ($$i\in I$$), the authors consider the problem of finding $$x\in K$$ such that $$F_{i}(x,y)\geq0$$ for all $$i\in I$$ and $$y\in K$$, as well as the problem of finding the projection of $$a\in\mathcal{H}$$ on $$S$$, the solution set of the preceding problem. In order to accomplish these aims, proximal-like block-iterative algorithms, as well as regularization and splitting algorithms, are proposed. For every algorithm, convergence results are established.

### MSC:

 90C48 Programming in abstract spaces 90C47 Minimax problems in mathematical programming 49K27 Optimality conditions for problems in abstract spaces